Infinity and Trinity

I've finished yet another reading of Infinity and the Mind by Rudy Rucker. As a mathematician, I always hear some strange ideas from people about infinity. Is it a number? Well, yes and no. It is not a number in the sense of standard arithmetic. In standard arithmetic, numbers can be added, subtracted, multiplied, and divided providing we are not dividing by zero. Share four cakes between two people and each person gets two cakes; i.e. 4÷2=2. Share four cakes between no people, how much does each person get? Dividing by zero is not technically infinity - it is technically meaningless.

Infinity is sort of a number when it comes to counting things. The trouble is that there are different types of infinity. Indeed there are bigger and bigger and bigger infinities until one reaches the Absolute Infinity - except we never reach the Absolute Infinity. All this is mathematically sound.

Now how do we understand infinities? We can say that they are the number of objects in a set. For the non-mathematician, a set is just a collection of specific and identifiable objects. We can call the set of numbers {1,2,3,4...} the set N for example. I call the set of all numbers which can be expressed as a decimal (which may be infinitely long,  like pi), R. It turns out that there are more numbers in R than in N.

Where on earth can this arcane discussion be going? As Rucker points out, a set is a Many that can be thought of as a One. Rather than think of all decimals, one by one, we can just conceive of R and use that to prevent ourselves from wasting intellectual resources.

The point is, we comprehend infinities by comprehending sets. Sets are the key thing here. You might think we can form a set of anything such as the set of all cats in Hull in 2016, or the set of bishop's wives - you are the set of all the cells in your body! Yet there are sets that cannot exist as sets such as the set S of all sets which don't contain themselves. That's a tricky concept and you can take time to think about it and enjoy the logical conundrum, or move on and just hold to my conclusion. There is no universal set, no set of all sets, but there is a class of all sets.

The point is that we can easily talk of the class of all sets as one object, in which case it really ought to be a set, but it can't be a set without some awful logical contradiction taking place. We can even conceive of T the class of all possible thoughts and the same thing happens. It is not a set. If a set is a Many that can be treated as a One, T is not. It is a single conceivable thing which cannot be treated as a single conceivable thing.

For the Christian, this sounds very familiar. We hold to a Unity that is a Multiplicity - a Three-in-One. Of course this is impossible by anything bound by the laws of Physics, but perhaps we see that in the laws of mathematics such a being can be possible albeit beyond finite comprehension.

I do apologise if I lost my readers along the way here. My aim was simply to show that the Doctrine of the Holy Trinity, although utterly transcendent, permits itself to be seen as a shadow in mathematical thought. Beyond that we must keep silence and lift ourselves above such Earthly thoughts.